A high‐order and fast scheme with variable time steps for the time‐fractional Black‐Scholes equation

Abstract

In this paper, a high‐order and fast numerical method is investigated for the time‐fractional Black‐Scholes equation. In order to deal with the typical weak initial singularity of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum‐of‐exponentials (SOE) technique. In the spatial direction, an average approximation with fourth‐order accuracy is employed. The stability and the convergence with second order in time and fourth order in space of the proposed scheme are religiously derived by the energy method. Numerical examples are given to demonstrate the theoretical statement.